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On Infinite Variants of NP-Complete Problems

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On Infinite Variants of NP-Complete Problems Journal of Computer and System Sciences SS1432 journal of computer and system sciences 53, 180193 (1996) article no. 0060 View metadata, citation and similarTaking papers at core.ac.uk It to the Limit: On Infinite Variants brought to you by CORE of NP-Complete Problems provided by Elsevier - Publisher Connector Tirza Hirst and David Harel* Department of Applied Mathematics H Computer Science, The Weizmann Institute of Science, Rehovot 76100, Israel Received November 1, 1993 We first set up a general way of obtaining infinite versions We define infinite, recursive versions of NP optimization problems. of NP maximization and minimization problems. If P is the For example, MAX CLIQUE becomes the question of whether a recursive problem that asks for a maximum by graph contains an infinite clique. The present paper was motivated by trying to understand what makes some NP problems highly max |[wÄ : A<,(wÄ , S)]|, undecidable in the infinite case, while others remain on low levels of S the arithmetical hierarchy. We prove two results; one enables using knowledge about the infinite case to yield implications to the finite where ,(wÄ , S) is first-order and A is a finite structure (say, case, and the other enables implications in the other direction. a graph), we define P as the problem that asks whether Moreover, taken together, the two results provide a method for proving there is an S, such that the set [wÄ : A<,(w, S)] is infinite, (finitary) problems to be outside the syntactic class MAX NP, and, where A is an infinite recursive structure. Thus, for example, hence, outside MAX SNP too, by showing that their infinite versions are 1 Max Clique becomes the question of whether a recursive 71-complete. We illustrate the technique with many examples, resulting 1 in a large number of new 71 -complete problems. ] 1996 Academic graph contains an infinite clique. Often, the infinitary Press, Inc. problem becomes trivial; the generalization of Max Sat, for instance, becomes the following problem: Given an infinite set of clauses C, does there exist some assignment S that satisfies an infinite subset of C? The answer is always in the 1. INTRODUCTION affirmative, when we consider only satisfiable clauses. In this paper we prove two results. One enables using An infinite recursive graph can be thought of simply as a information about the infinite case to yield implications to recursive binary relation over some recursive countable the finite case, and the other enables implications in the setthe natural numbers, for instance. Since recursive other direction. Moreover, taken together, the two results graphs can be represented by the Turing machines that provide a method for proving (finitary) problems to be out- recognize their edge sets, one can investigate the complexity side the syntactic class Max NP, and, hence, outside Max of problems concerning them. Indeed, a significant amount SNP too,2 while at the same time showing infinitary of work has been carried out in recent years regarding such 1 Max Max problems to be 71 -hard. The classes NP and problems. Beigel and Gasarch [BG1, BG2] have shown SNP have been the subject of recent renewed interest, that many problems on recursive graphs reside on low levels following the developments that show that, whereas all of the arithmetical hierarchy. For example, determining problems in Max NP are approximable in polynomial time whether a recursive graph is 3-colorable is 6 0 -complete. On 1 to within some constant, the ones that are hard for Max the other hand, in [H2] it was shown that determining SNP have no polynomial-time approximation scheme whether a recursive graph has a Hamiltonian path is outside [ALMSS, AS]. These latter sets are closed under special the arithmetical hierarchy, and is, in fact, 71 -complete.1 1 approximation-preserving transformations and are, thus, The present work was motivated by trying to understand different from the syntactic sets we refer to here. In fact, our what makes some NP-complete problems on graphs highly results appear to have no direct relevance to issues of undecidable in the infinite case, while others remain on low approximability. levels of the arithmetical hierarchy. Our first result, proved in Section 3, is that the infinite version of any problem in Max NP is arithmetical; specifi- Max 0 Max * E-mail: [tirza, harel]Äwisdom.weizmann.ac.IL. cally, if P # NP then P # 6 2 . Moreover, if P # 1 A similar result for perfect matching in recursive graphs can be established using a proof technique appearing in [AMS]. 2 These classes, which appear in [PY, PR, KT], are defined in Section 2. 0022-0000Â96 18.00 180 Copyright 1996 by Academic Press, Inc. All rights of reproduction in any form reserved. File: 571J 143201 . By:BV . Date:25:09:96 . Time:14:41 LOP8M. V8.0. Page 01:01 Codes: 6866 Signs: 4588 . Length: 60 pic 11 pts, 257 mm INFINITE VARIANTS OF NP-COMPLETE PROBLEMS 181 NP then every instance of P has either an infinite solution for quantifier-free . Max Sat is the canonical example of a or a maximal finite solution (i.e., no instance of P has a problem in Max NP. They also considered the subclass solution of size k, for arbitrarily large k, without having an Max SNP of Max NP, consisting of those maximization infinite one too). We like to view this fact in its dual problems that are defined by quantifier-free formulas, i.e, formulation: The finitary version of any problem whose the optimum of such problems can be defined by 0 Max infinite version is higher than 6 2 must be outside NP, and, hence, outside Max SNP too. The same is true for max |[xÄ : A<(xÄ , S)]| problems that have an instance containing no infinite solu- S tion, but containing a solution of size k, for arbitrarily large k. For the second result, proved in Section 4, we define for quantifier-free . Max 3SAT is easily seen to be in Max a special kind of monotonic transformation between NP SNP. Actually, Papadimitriou and Yannakakis [PY] optimization problems, which we call an M-reduction. The meant that the classes Max NP and Max SNP contain also idea is, essentially, that (in a maximization problem) enriching their closures under L-reductions, which preserve polyno- the structure in one problem enriches it in the other, as well mial-time approximation schems. We do not. Kolaitis and Max Max Max as making the objective functions grow. We prove that M- Thakur use the names 70 and 71 , for SNP reductions between conventional finitary problems become and Max NP, and they too talk about the ``pure'' syntactic 1 71 -reductions when ``lifted up'' to the infinite case. This classes. To avoid confusion, we shall follow this terminology 1 enables one to prove 71 -hardness of infinitary problems by in the rest of the paper. examining, and sometimes modifying, reductions between More recently, in a paper by Panconesi and Ranjan their finitary versions. [PR], Kozen showed that Max Clique does not belong to Max Max Indeed, in Section 5 we use our second result to prove the 71 . 61 was introduced in [PR] as the class of 1 71 -hardness of many additional problems. Moreover, by maximization problems whose optimum can be defined by our first result, the finitary versions of these must all be out- side Max NP. Here is a partial list of the problems for which max |[wÄ : A<(\xÄ ) (wÄ , xÄ , S)]| these two properties are established: Max Clique, Max S Ind Set, Max Ham Path, Max Subgraph, Max Common Subsequence, Max Color, Max Exact Cover By Pairs, for quantifier-free . Max Tiling. Kolaitis and Thakur [KT] then took a broader view. They examined the class of all maximization problems 2. BACKGROUND AND PRELIMINARIES whose optimum is definable using first-order formulas, i.e., by Our approach to optimization problems is to focus on max |[wÄ : A<(wÄ , S)]|, their descriptive complexity, via logical definability, an idea S that started with Fagin's [F] characterization of NP in terms of definability in second-order logic on finite struc- where (wÄ , S) is an arbitrary first-order formula. They first tures. (The following paragraphs are adapted from [KT].) showed that this class coincides with the collection of poly- An existential second-order formula is an expression of the nomially bounded NP-maximization problems on finite form (_S) ,(S), where S is a sequence of second-order structures, i.e., those problems whose optimum value is variables that can contain relations (predicates), and ,(S)is bounded by a polynomial in the input size. They then a first-order formula over some vocabulary _. The formula proceeded to show that these problems form a proper is finitary, so the number of variables in S, xÄ , and yÄ is some hierarchy, with exactly four levels: fixed finite constant. Fagin's theorem [F] asserts that a collection C of finite structures over some vocabulary _ is Max 7 /Max 7 /Max 6 /Max 6 = . Max 6 . NP-computable if and only if there is a quantifier-free 0 1 1 2 i i2 formula (xÄ , yÄ , S ) over _, such that for any finite structure Max Max Max A we have Here, 61 is defined just like 71 (i.e., NP), Max but with a universal quantifier, and 62 uses a universal A # C A<(_S)(\xÄ )(_yÄ ) (xÄ , yÄ , S ). followed by an existential quantifier, and corresponds to Fagin's general result stated above. The three containments Papadimitriou and Yannakakis [PY] introduced the class are strict: It is shown in [KT] that Max Connected Max Component Max Max Max NP of maximization problems, whose optimum can be is in 62 but not in 61 , while Clique Max Max defined by is in 61 but not in 71 (the latter fact was mentioned above and appears in [PR]), and Max Sat is in max |[xÄ : A<(_yÄ ) ( xÄ , yÄ , S ) ] |, Max Max S 71 but not in 70 (this is from [PY]).
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